Floquet Engineering in Quantum Chains

We consider a one-dimensional interacting spinless fermion model, which displays the well-known Luttinger liquid (LL) to charge density wave (CDW) transition as a function of the ratio between the strength of the interaction $U$ and the hopping $J$. We subject this system to a spatially uniform drive which is ramped up over a finite time interval and becomes time periodic in the long-time limit. We show that by using a density matrix renormalization group approach formulated for infinite system sizes, we can access the large-time limit even when the drive induces finite heating. When both the initial and long-time states are in the gapless (LL) phase, the final state has power-law correlations for all ramp speeds. However, when the initial and final state are gapped (CDW phase), we find a pseudothermal state with an effective temperature that depends on the ramp rate, both for the Magnus regime in which the drive frequency is very large compared to other scales in the system and in the opposite limit where the drive frequency is less than the gap. Remarkably, quantum defects (instantons) appear when the drive tunes the system through the quantum critical point, in a realization of the Kibble-Zurek mechanism.

Figure 1

Ground state phase diagram of Eq. (1) as a function of ratio of interaction strength $U$ to hopping $J$ showing Luttinger liquid (LL) and charge density wave (CDW) phases. Notations (a1)–(a3) and (b) indicate the studied cases, with the direction in which the effective interaction strength can be tuned by a nonequilibrium drive indicated by the arrow head. The circular marker signals the location of the LL to CDW quantum phase transition.

Figure 2

Density-density correlation function $C(j)$ averaged over the time range $t=100/J$ to $t=400/J$ for different $j$ (solid lines) as function of inverse of ramp time $\tau$ for $\mathrm{\Omega }/J=10$, $E_0/\mathrm{\Omega }=1$, and $T/J=0$ (Magnus limit). Top: Initial correlation strength $U/J=0.5$ (LL phase), Magnus estimate of final correlation strength $U/J_{\mathrm{eff}}\approx 0.65$. Middle: Initial state $U/J=1.75$ (CDW), final $U/J_{\mathrm{eff}}\approx 2.29$ (CDW). Bottom: Initial state $U/J=4$ (CDW), final $U/J_{\mathrm{eff}}\approx 5.23$ (CDW). The ground state expectation values from the effective Hamiltonian ($J\to J^{\mathrm{eff}}$) are given as horizontal dashed lines and the gaps ${\mathrm{\Delta }}^{\mathrm{eff}}$ as vertical dashed lines.

Figure 3

Main: Time-averaged density-density correlation function $C(j)$ (solid lines) for different $J\tau$ and $U/J$. The other parameters are $\mathrm{\Omega }/J=10$, $F_0/\mathrm{\Omega }=1$, and $T/J=0$. The $T=0$ equilibrium prediction with $J\to J^{\mathrm{eff}}=J_0(E_0/\mathrm{\Omega })J$ (dark dashed lines) compares well to the time-averaged results only in the limit of large $J\tau$. The light dashed lines are the undriven equilibrium results. Inset: Effective temperature (crosses) as function of $\tau$ extracted by fitting the slope of the exponential tail of the $U/J=1.7$ and $U/J=4$ lines in the main panel. One of these fits ($U/J=1.7$ and $J\tau =0.5$) of the slope of the exponential tail in $j$ is shown in the main panel by open circles. The approximate prediction $T^{\mathrm{eff}}/{\mathrm{\Delta }}^{\mathrm{eff}}=a\mathrm{\Delta }\tau \sinh (b\tau \mathrm{\Delta })$ is the dashed line (see SM [34]).

Figure 4

Main: Time-averaged density-density correlation function $C(j)$ (lines) for different $E_0/\mathrm{\Omega }$. The other parameters are $J\tau =4$, $\mathrm{\Omega }/U=0.6$, $U/J=16$, and $T/J=0$. Inset: Comparison of Eq. (3) for effective hopping generated by subgap drive (solid line) and deduced from fits of the data shown in the main panel to the equilibrium $T-0$ formula for the long-distance limit of $C$.

Figure 5

Main: The correlation function $C(j)$ at different times averaged over one drive period for the Kibble-Zurek (a3) case. Lines are shifted vertically for clarity of depiction; the midpoint of the oscillation is zero. The dashed black line gives the asymptotic value expected at large $j$ from a ground state calculation using the Magnus formalism, shifted to correspond to the longest time case. Inset: Time evolution of the phase averaged over the drive period. Light and dark gray denote the phase of $C+1$ and $-1$ with respect to the small $j$ oscillation, respectively. The parameters are $U/J=0.95$, $E_0/\mathrm{\Omega }=1.5$, $J\tau =4$, and $T/J=0$.